conference papers
860 # 2003 International Union of Crystallography
Printed in Great Britain ± all rights reserved J. Appl. Cryst. (2003). 36, 860±864
Uniqueness of ab initio shape determination in
small-angle scattering
Vladimir V. Volkov
a*
and Dmitri I. Svergun
b,a
a
Institute of Crystallography, Russian Academy of Sciences,
Moscow, Russia, and
b
European Molecular Biology Laboratory,
c/o DESY, Hamburg, Germany. E-mail: vvo@ns.crys.ras.ru
Scattering patterns from geometrical bodies with different shapes
and anisometry (solid and hollow spheres, cylinders, prisms) are
computed and the shapes are reconstructed ab initio using envelope
function and bead modelling methods. A procedure is described to
analyze multiple solutions provided by bead modeling methods and
to estimate stability and reliability of the shape reconstruction. It is
demonstrated that flat shapes are more difficult to restore than
elongated ones and types of shapes are indicated, which require
additional information for reliable shape reconsrtuction from the
scattering data.
Keywords: Small-angle scattering; shape determination;
uniqueness; structure modeling
1. Introduction
Small-angle scattering (SAS) patterns from monodisperse systems of
non-interacting particles (e.g. from dilute protein solutions) are
isotropic functions I(s) of the momentum transfer s = 4πsin
θ
/
λ
,
where 2
θ
is the scatting angle and
λ
is the radiation wavelength. Due
to chaotic positions of individual particles, I(s) is proportional to the
scattering from a single particle averaged over all orientations
(Feigin & Svergun, 1987). Ab initio analysis aiming at recovering
three-dimensional structure from one-dimensional scattering curve is
obviously ambiguous, as many different models may yield the same
SAS curve with near the same accuracy. Homogeneous
approximation is often used to constrain the solution by assuming
uniform density inside the particle and discarding the influence of its
internal structure. Such a simplification can be employed e.g. in the
analysis of low resolution (to about s=2-3 nm
-1
) portions of X-ray
scattering patterns from sufficiently large (>50 KDa) proteins.
In the past, trial-and-error method was employed for shape
modeling by computing the scattering patterns from different shapes
to compare them with the experimental data. One could distinguish
between two modeling strategies. One strategy was to keep the
number of model parameters as low as possible, using three-
parameter bodies like prisms, ellipsoids or cylinders; the other was
to construct complicated bodies from collections of spheres (i.e. to
use many parameters) and to restrain the model by additional
information (Kratky & Pilz, 1978). Evolution of the two strategies
and improved power of computers lead to the modern ab initio shape
determination methods. These methods are now widely employed in
practice by different research groups, e.g. (Bada et al., 2000;
Grossmann et al., 2000; Bernocco et al., 2001; Egea et al., 2001;
Scott et al., 2002) but the question of uniqueness of the models is
rarely discussed. In the present paper, the problem of uniqueness of
ab initio shape analysis is addressed by model calculations on
particles of different shape and anisometry, and practical
recommendations are given for assessing the reliability of the shape
reconstruction.
2. Shape determination methods
In the first general ab initio method based on a few parameters
approach, Stuhrmann (1970) proposed to represent the particle shape
by an angular envelope function r=F(
ω
), where (r,
ω
) are spherical
coordinates. The particle density was unity inside the envelope and
zero outside. The envelope was described by a series of spherical
harmonics:
)()(
0
ωω
lmlm
l
lm
L
l
YfF
∑∑
−==
= (1)
where the maximum order of harmonic L defined the resolution. The
low resolution shape is thus defined in a general case by a (L+1)
2
-6
parameters, which can determined by a non-linear minimization
procedure to fit the scattering data. This approach was further
developed by (Svergun & Stuhrmann, 1991) who proposed
algorithms for rapid computation of scattering intensities from such
a model and implemented in the computer program SASHA
(Svergun et al., 1996). It was demonstrated that in practice a unique
envelope can be extracted from the scattering data up to the
resolution L=4.
The modeling using angular envelope function has limitations in
describing complicated, e.g. very anisometric, particles, or those
having internal cavities. Although the solution is unique (of course,
up to an enantiomorphic shape, which always provides the same
intensity), series (1) may not ensure adequate representation of the
shape leading to systematic errors (even if the scattering data is
neatly fitted). A more comprehensive description is achieved in the
bead modeling methods, which use the improved speed of modern
computers to revive the strategy of many parameter modeling in
different flavors of Monte Carlo-type search. The ab initio bead
modeling in a confined volume was first proposed by (Chacon et al.,
1998; Chacon et al., 2000). The maximum dimension D
max
of
a
particle is readily obtained from the scattering pattern and the
particle must obviously fit inside a sphere of this diameter. If one
fills the sphere with M densely packed beads (spheres of radius r
0
<< D
max
), each of these beads may belong either to the particle (X
i
=1) or to the solvent (X
i
=0), and the particle shape is described by a
string, X, of M bits. Scattering intensity from the bead model is
computed e.g. using Debye’s formula (Debye, 1915)
ij
ij
M
j
ji
M
i
sr
sr
XXssI
)sin(
)(f)(
11
2
∑∑
==
= (2)
where r
ij
= |r
i
-r
j
| is the distance between the beads and f(s) is the bead
form factor (scattering amplitude from a sphere of radius r
0
).
Starting from a random distribution of 1 and 0, the model is
modified to find the binary string (i.e. the shape) that fits the
experimental data using a genetic algorithm. In a more general
approach (Svergun, 1999), the beads may belong to different
components so that the shape and internal structure o
f
multicomponent particles (e.g. nucleoproteins) can be reconstructed
by simultaneously fitting scattering data at different contrasts
(Svergun & Nierhaus, 2000). For single component particles, the
procedure degenerates to an ab initio shape determination. The
model intensity is computed using spherical harmonics to speed up
the computations and compactness and connectivity constraints are
imposed in the search, implemented in the simulated annealing
program DAMMIN (Svergun, 1999). Ab initio Monte Carlo-type
approaches without limitation of the search space are also available
(program SAX3D, (Walther, Cohen & Doniach, 2000) and the
program SASMODEL (Vigil et al., 2001)).
J. Appl. Cryst. (2003). 36, 860±864 Volkov and Svergun 861
conference papers
Figure 1
Scattering curves computed from the model bodies (circles) and fits by
SASHA (dashed lines) and DAMMIN (solid lines). The numbers correspond
to those in Figs. 2-5. The abscissae and ordinates of the individual curves are
multiplied by appropriate scale factors for better visualization.
Figure 2
Shape determination of globular solid particles. Here and below, the models
(from left to right) are: the geometrical body to be restored; SASHA envelope
model (if applicable); one or two typical DAMMIN models; the two
rightmost panels always display the results of DAMAVER: TSR (left) an
d
MPV (right). The models (except those with spherical shape) are displayed in
two orthogonal views. All bodies are marked by successive numbers (left
column) through Fig. 2 to 5. Structure proportions are indicated at the top o
f
each geometrical body as ratio of diameter to hight (solid cylinders), ratio o
f
edges (prisms), ratio of inner and outer diameters (hollow spheres), ratio
inner diameter - outer diameter - hight (hollow cylinders).
The search models employed in all Monte-Carlo base
d
approaches are described by hundreds or thousands parameters (e.g.
occupancy indices in bead modelling). Running these programs
several times on the same data starting from different initial
approximations may yield different final models, and the question
arises, how reliable is the many-parameter approach. The authors o
f
the shape determination programs usually present their successfu
l
applications to simulated and practical examples. The present pape
r
is an attempt to explore limits of ab initio shape determination by
performing model calculations on geometrical bodies to find out
which shapes and structural details can and which cannot be reliably
determined from the scattering data.
3. Model calculations
The geometrical bodies taken for the simulations differed by shape
(spheres, cylinders and prisms) and anisometry. Elongated and
flattened particles (ratio of length to width 1:C and C:1, respectively,
1<C<10) and hollow particles with the ratio of the inner radius to the
outer radius 0.33 < r/R < 0.66 were considered. Scattering patterns
conference papers
862 Volkov and Svergun J. Appl. Cryst. (2003). 36, 860±864
were calculated using analytical equations (Feigin & Svergun, 1987)
in the angular range covering about 15 to 18 Shannon channels (the
width of a Shannon cnannel is
∆
s=π/D
max
(Shannon & Weaver,
1949)). This ensured approximately the same information content in
the simulated data for different shapes (Moore, 1980). The computed
curves were used to reconstruct the shape ab initio using the
simulated annealing program DAMMIN (Svergun, 1999), and, for
the shapes without cavities, also using the envelope program
SASHA (Svergun et al., 1996). The results provided by the ab initio
programs were compared to the actual shapes. Both programs were
run in batch modes using default answers without symmetry
restrictions. For SASHA, multipole resolutiomn of L=6 was used; all
DAMMIN calculations were made inside the search volume with
diameter 1.05*D
max
; 'fast' mode was used for globular particles and
'slow' mode for anisometric particles.
Figure 3
Shape determination of hollow globular particles. See annotation to Fig. 2 for
details.
The envelope determination technique (program SASHA) gives
a single solution. In contrast, DAMMIN provides many solutions
(spatial distributions of beads) for runs with different seeds for
random number generator (i.e. with randomly generated starting
models). Analysis of the DAMMIN solutions yielding nearly
identical scattering patterns can serve as an indicator of the stability
of the solution. For automated analysis of independent DAMMIN
reconstructions, a program package DAMAVER was written based
on the program SUPCOMB (Kozin & Svergun, 2001). The latter
program aligns two arbitrary low or high resolution models
represented by ensembles of points by minimizing a dissimilarity
measure called normalized spatial discrepancy (NSD). For every
point (bead or atom) in the first model, the minimum value among
the distances between this point and all points in the second model is
found, and the same is done for the points in the second model.
These distances are added and normalized against the average
distances between the neighboring points for the two models.
Generally, NSD values close to one indicate that the two models are
similar.
Figure 4
Shape determination of anisometric solid particles. See annotation to Fig. 2
for details.
For each model body, ten independent DAMMIN
reconstructions were analyzed by DAMAVER as follows. The
values of NSD were computed between each pair in the set and a
mean value over all pairs <NSD> and dispersion
∆
(NSD) were
calculated. For each reconstruction, the average value NSD
k
with
respect to the rest of the set was computed and the reference
reconstruction with lowest NSD
k
was selected. Possible outliers with
NSD
k
exceeding <NSD> + 2
∆
(NSD) were discarded. All the
models except the outliers were superimposed onto the reference
model using SUPCOMB and the entire assembly of beads was
remapped onto a densely packed grid of beads where each grid point
was characterized by its occupancy factor (the number of beads in
the entire assembly that are in the vicinity of the grid point). The grid
points with non-zero occupancy form a total spread region (TSR),
J. Appl. Cryst. (2003). 36, 860±864 Volkov and Svergun 863
conference papers
and a portion of the TSR with higher occupancies was selected (most
populated volume, MPV) to yield the volume equal to the average
excluded volume of all the reconstructions. The scattering computed
from the MPV would not fit the experimental data but this model
should preserve the most probable features of the solution.
Figure 5
Shape tetermination of hollow anisometric and acentric particles. See
annotation to Fig. 2 for details.
4. Results
The scattering patterns from the model bodies are presented in Fig. 1
along with the fits provided by the ab initio reconstructions (which
were in most cases undistinguishable from the theoretical scattering
patterns).
Fig. 2 demonstrates that solid bodies with moderate anisometry
(elongated particles up to 1:5 and flattened up to 5:2) can be reliably
reconstructed from the scattering data. The shapes obtained by
SASHA reasonably represent the overall anisometry, albeit
displaying artificial features due to limited resolution of series (1).
DAMMIN yields very stable reconstructions, which is reflected in
the mean value <NSD> of about 0.4-0.7 for all these cases. Hollow
globular models can also be well reconstructed (Fig. 3). For a hollow
concentric sphere, even rather small voids with r/R = 0.33, are
clearly revealed, and the shape of a hollow cylinder with r/R = 0.5 is
also neatly restored.
Shape reconstructions of anisometric particles are less stable an
d
reliable. For elongated bodies, anisometry 1:5 is limiting for SASHA
whereas DAMMIN still represents an elongated particle with the
ratio 1:10, but tends to provide a slightly bent shape, even after the
averaging procedure (Fig. 4, bodies 9 and 10 , <NSD> = 0.5 and 0.6,
respectively). Flattened particles represent yet more difficult case,
and starting from the anisometry 5:1, the shapes provided by
SASHA are meaningless. The individual solutions from DAMMIN
also show artifacts but for the anisometry 5:1 (Fig. 4, body 11),
<NSD> = 0.75) the MPV reasonably well represents the flat initial
shape. For the anisometry 10:1, the TSR is very large (<NSD> = 1.3)
and even the MPV does not resemble a disk-shaped particle (Fig. 4,
body 12).
Elongated hollow particles with higher anisometry or narrow
channels (Fig. 5, bodies 13,14) may also pose problems for the shape
reconstruction. Although the DAMMIN solutions are stable (<NSD>
about 0.45-0.65), the channels may appear closed from one or both
sides, in individual solutions and also in the MPV. Similar to what
was observed for solid models, hollow flattened particles are even
more difficult to restore: for different r/R ratios, the resulting shapes
may show a helical turn instead of a hollow disk, even after the
averaging (Fig. 5, bodies 15,16). Acentric voids in hollow spheres
are only reconstructed if r/R is about 0.5 (Fig. 5, body 17); smaller
voids are just "dispersed" inside the entire spherical volume (Fig. 5,
body 18). For globular particles with small cavities, the <NSD>
value may be small, (typically 0.4-0.6) indicating overall similarity
of the models, but the details of the internal structure may
significantly differ between the solutions.
5. Discussion
In the present study, we tried to find out simple geometrical shapes,
which cannot be reliably restored from the small-angle scattering
data. That is why we used relatively wide ranges of the scattering
vectors and did not add noise to the simulated profiles. If a shape
cannot be reliably restored under ideal conditions, it is unrealistic to
hope and restore it ab initio from real experimental data without
additional information. The bodies with sharp edges (cylinders and
prisms) were deliberately taken instead of ellipsoids, because the
edges are generally more difficult to reconstruct.
We have performed extensive computations on geometrical
bodies with different shapes, anisometries and internal cavities; this
paper presents a selection of most representative results. The main
conclusions from the model calculations are:
(i) Flattened particles are generally more difficult to reconstruct
than the elongated ones. The degree of anisometry for a reliable ab
initio shape reconstruction should not exceed 1:10 and 5:1 for
elongated and flattened particles, respectively. Not surprisingly, it
was found that for anisometric particles the bead modeling is
superior over the envelope determination technique. In general, it is
advisable to check the particle anisometry, e.g. by finding a best-fit
three-axial ellipsoid - the option provided, in particular, in the
program SASHA - prior to running a shape determination program.
conference papers
864 Received 29 August 2002
Accepted 2 January 2003 J. Appl. Cryst. (2003). 36, 860±864
(ii) For globular and elongated particles, internal cavities as
small as 1/3 of diameter or cross-section can be restored. For flat
hollow particles, the general shape is reconstructed but artifacts may
appear (e.g a disk may evolve to a helical turn). For spherical
particles, concentric voids are better reconstructed than acentric
ones.
(iii) The mean NSD between independent reconstructions
obtained by bead modeling provide a useful estimate of the
reliability of the solution. The values of <NSD> exceeding 0.7 yield
large TSR and indicate that the reconstruction is unstable; in these
cases additional information is required for reliable shape
determination.
(iv) Averaging of the independent reconstructions allows one to
enhance the most persistent features of the bead models and in most
cases improves the quality of the shape reconstruction. There could,
however, be cases, when the averaging has little effect and the MPV
still shows systematic deviations from the initial shape (Fig. 2, body
2; Fig. 5, bodies 15,16).
The programs SASHA and DAMMIN were used for the
simulations because they were written by the authors. Calculations
with the programs DALAI_GA (Chacon et al., 1998; Chacon et al.,
2000) and SAXS3D (Walther, Cohen & Doniach, 2000) yielded the
results similar to DAMMIN (somewhat less stable, but this might be
attributed to not necessarily optimal choice of parameters). All
computations in the present paper were made without imposing
symmetry restrictions on possible models, and without information
about particle anisometry (in particular, for DAMMIN, within
spherical search volumes. Although both SASHA and DAMMIN
allow one to impose restrictions both on symmetry and anisometry,
these restrictions were not used to keep the results as general as
possible. It must thus be stressed that our conclusions refer to "pure"
ab initio shape determination without any additional information.
The use of symmetry allows one to reliably restore even highly
flattened particles, (see e.g. model calculations by Volkov et. al., this
issue).
The programs SASHA and DAMMIN, along with the averaging
package DAMAVER, are available from www.embl-
hamburg.de/ExternalInfo/Research/Sax. The program DAMAVER
can be used to analyse the stability and construct average models
provided by different ab initio methods, in particular also by the
dummy residues method of (Svergun, Petoukhov & Koch, 2001)
The authors acknowledge the financial support provided by the
Russian Foundation for Basic Research, grant No 01-02-17040, and
by the INTAS grant No 00-243.
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